In the men's singles, Parupalli Kashyap cruised into the second round, beating Rajiv Ouseph of England 21-15, 21-18.
Saina had to work hard for a 12-21, 21-10, 21-12 victory over Karin
Schnaase of Germany, while Sindhu needed just 39 minutes to get the better of Hong Kong's Pui Yin Yip 21-13, 22-20 at the Odense Sports Park.
London Olympics bronze medallist Saina, ranked No 7, will next take on the winner of the match between Li Han and Minatsu Mitani.
But for a late fightback from Yin Yip, world No 10 Sindhu had things her way before advancing.
She will face the winner of the first round tie between Mette Poulsen and Ksenia Polikarpova.
Earlier, Ashwini Ponnappa and Russia's Vladimir Ivanov lost 18-21, 18-21 in 30 minutes to the Hong Kong pair of Yun Lung Chan and Ying Suet Tse.
Later in the day, Kidambi Srikanth registered a 21-15, 17-21, 21-18 victory over China's Xue Song in a match that lasted one hour and 11 minutes.
But the doubles specialist pairing of Jwala Gutta and Ashwini Ponnappa were beaten in straight games by the Dutch duo of Eefje Muskens and Selena Piek 21-17, 21-15.
Image: Saina Nehwal
Countries with most millionaires: INDIA ranks 11
Twitter wants to don a more sporty avatar in India
Sports Shorts: Saina progresses; Benzema, Ronaldo on target for Real
India's C'wealth Games showing raises hope for big medal haul at Asiad
Saina won't train with Gopichand at Asiad, asks BAI to include Vimal